Fourier series: a modern introduction/
R. E. Edwards
- 2nd. ed.
- New York : Springer, 1982.
- 2 v. (369p.) : 24cm.
- Graduate texts in mathematics, 85. .
11 Spans of Translates. Closed Ideals. Closed Subalgebras. Banach Algebras -- 11.1 Closed Invariant Subspaces and Closed Ideals -- 11.2 The Structure of Closed Ideals and Related Topics -- 11.3 Closed Subalgebras -- 11.4 Banach Algebras and Their Applications -- Exercises -- 12 Distributions and Measures -- 12.1 Concerning C? -- 12.2 Definition and Examples of Distributions and Measures -- 12.3 Convergence of Distributions -- 12.4 Differentiation of Distributions -- 12.5 Fourier Coefficients and Fourier Series of Distributions -- 12.6 Convolutions of Distributions -- 12.7 More about M and Lp -- 12.8 Hilbert's Distribution and Conjugate Series -- 12.9 The Theorem of Marcel Riesz -- 12.10 Mean Convergence of Fourier Series in LP (1 12.11 Pseudomeasures and Their Applications -- 12.12 Capacities and Beurling's Problem -- 12.13 The Dual Form of Bochner's Theorem -- Exercises -- 13 Interpolation Theorems -- 13.1 Measure Spaces -- 13.2 Operators of Type (p, q) -- 13.3 The Three Lines Theorem -- 13.4 The Riesz-Thorin Theorem -- 13.5 The Theorem of Hausdorff-Young -- 13.6 An Inequality of W.H. Young -- 13.7 Operators of Weak Type -- 13.8 The Marcinkiewicz Interpolation Theorem -- 13.9 Application to Conjugate Functions -- 13.10 Concerning?*f and s*f -- 13.11 Theorems of Hardy and Littlewood, Marcinkiewicz and Zygmund -- Exercises -- 14 Changing Signs of Fourier Coefficients -- 14.1 Harmonic Analysis on the Cantor Group -- 14.2 Rademacher Series Convergent in L2 -- 14.3 Applications to Fourier Series -- 14.4 Comments on the Hausdorff-Young Theorem and Its Dual -- 14.5 A Look at Some Dual Results and Generalizations -- Exercises -- 15 Lacunary Fourier Series -- 15.1 Introduction of Sidon Sets -- 15.2 Construction and Examples of Sidon Sets -- 15.3 Further Inequalities Involving Sidon Sets -- 15.4 Counterexamples concerning the Parseval Formula and Hausdorff-Young Inequalities -- 15.5 Sets of Type (p, q) and of Type?(p) -- 15.6 Pointwise Convergence and Related Matters -- 15.7 Dual Aspects: Helson Sets -- 15. 8 Other Species of Lacunarity -- Exercises -- 16 Multipliers -- 16.1 Preliminaries -- 16.2 Operators Commuting with Translations and Convolutions; m-operators -- 16.3 Representation Theorems for m-operators -- 16.4 Multipliers of Type (LP, Lq) -- 16.15 A Theorem of Kaczmarz -- Stein -- 16.6 Banach Algebras Applied to Multipliers -- 16.7 Further Developments -- 16.8 Direct Sum Decompositions and Idempotent Multipliers -- 16.9 Absolute Multipliers -- 16.10 Multipliers of Weak Type (p, p) --